Properties of the Shifted Graph

Oct 13, 2025 by windailys.com 145 views

Introduction

Logarithmic functions are a fundamental concept in mathematics, and their graphs play a crucial role in understanding various mathematical concepts. In this article, we will explore the graph of the logarithmic function $$g(x)=\log_4 x$$ and discuss how to obtain the graph of the function $$f(x)=6+\log_4(x)$$ by shifting the graph of $$g(x)$$.

Logarithmic Functions

A logarithmic function is a function that is the inverse of an exponential function. The general form of a logarithmic function is $$f(x)=\log_b x$$, where $$b$$ is the base of the logarithm. In this case, we have $$g(x)=\log_4 x$$, which means that the base of the logarithm is 4.

Graph of $$g(x)=\log_4 x$$

The graph of a logarithmic function is a curve that increases slowly at first and then more rapidly as the input value increases. The graph of $$g(x)=\log_4 x$$ is a specific example of a logarithmic function with a base of 4.

Properties of the Graph

The graph of $$g(x)=\log_4 x$$ has several important properties:

Domain: The domain of the function is all positive real numbers, i.e., $$x>0$$.
Range: The range of the function is all real numbers, i.e., $$y\in\mathbb{R}$$.
Asymptote: The graph has a vertical asymptote at $$x=0$$, which means that the function approaches infinity as $$x$$ approaches 0 from the right.
Intercepts: The graph has a y-intercept at $$(0, -\infty)$$, which means that the function approaches negative infinity as $$x$$ approaches 0 from the left.

Shifting the Graph

To obtain the graph of $$f(x)=6+\log_4(x)$$, we need to shift the graph of $$g(x)=\log_4 x$$ 6 units up. This means that we need to add 6 to the y-coordinate of each point on the graph of $$g(x)$$.

Properties of the Shifted Graph

The graph of $$f(x)=6+\log_4(x)$$ has several important properties:

Domain: The domain of the function is all positive real numbers, i.e., $$x>0$$.
Range: The range of the function is all real numbers greater than or equal to 6, i.e., $$y\geq 6$$.
Asymptote: The graph has a vertical asymptote at $$x=0$$, which means that the function approaches infinity as $$x$$ approaches 0 from the right.
Intercepts: The graph has a y-intercept at $$(0, 6)$$, which means that the function approaches 6 as $$x$$ approaches 0 from the left.

Conclusion

In conclusion, the graph of the function $$f(x)=6+\log_4(x)$$ can be obtained from the graph of $$g(x)=\log_4 x$$ by shifting the graph of $$g(x)$$ 6 units up. This means that we need to add 6 to the y-coordinate of each point on the graph of $$g(x)$$ to obtain the graph of $$f(x)$$. The graph of $$f(x)$$ has several important properties, including a domain of all positive real numbers, a range of all real numbers greater than or equal to 6, a vertical asymptote at $$x=0$$, and a y-intercept at $$(0, 6)$$.

Example Problems

Here are some example problems to help you practice working with logarithmic functions and graph transformations:

Problem 1: Find the value of $$\log_4 16$$.
Problem 2: Graph the function $$f(x)=2+\log_4(x)$$.
Problem 3: Find the value of $$\log_4 1$$.
Problem 4: Graph the function $$f(x)=\log_4(x-2)$$.

Solutions

Here are the solutions to the example problems:

Problem 1: $$\log_4 16=2$$
Problem 2: The graph of $$f(x)=2+\log_4(x)$$ is a shifted version of the graph of $$g(x)=\log_4 x$$, with a y-intercept at $$(0, 2)$$.
Problem 3: $$\log_4 1=0$$
Problem 4: The graph of $$f(x)=\log_4(x-2)$$ is a shifted version of the graph of $$g(x)=\log_4 x$$, with a horizontal asymptote at $$x=2$$.

Final Thoughts

Introduction

In our previous article, we discussed logarithmic functions and graph transformations. We explored the graph of the logarithmic function $$g(x)=\log_4 x$$ and discussed how to obtain the graph of the function $$f(x)=6+\log_4(x)$$ by shifting the graph of $$g(x)$$. In this article, we will answer some frequently asked questions about logarithmic functions and graph transformations.

Q&A

Q: What is a logarithmic function?

A: A logarithmic function is a function that is the inverse of an exponential function. The general form of a logarithmic function is $$f(x)=\log_b x$$, where $$b$$ is the base of the logarithm.

Q: What is the graph of a logarithmic function like?

A: The graph of a logarithmic function is a curve that increases slowly at first and then more rapidly as the input value increases. The graph of $$g(x)=\log_4 x$$ is a specific example of a logarithmic function with a base of 4.

Q: What are the properties of the graph of a logarithmic function?

A: The graph of a logarithmic function has several important properties:

Domain: The domain of the function is all positive real numbers, i.e., $$x>0$$.
Range: The range of the function is all real numbers, i.e., $$y\in\mathbb{R}$$.
Asymptote: The graph has a vertical asymptote at $$x=0$$, which means that the function approaches infinity as $$x$$ approaches 0 from the right.
Intercepts: The graph has a y-intercept at $$(0, -\infty)$$, which means that the function approaches negative infinity as $$x$$ approaches 0 from the left.

Q: How do I shift the graph of a logarithmic function?

A: To shift the graph of a logarithmic function, you need to add a constant to the y-coordinate of each point on the graph. For example, to shift the graph of $$g(x)=\log_4 x$$ 6 units up, you need to add 6 to the y-coordinate of each point on the graph to obtain the graph of $$f(x)=6+\log_4(x)$$.

Q: What are the properties of the shifted graph?

A: The graph of $$f(x)=6+\log_4(x)$$ has several important properties:

Domain: The domain of the function is all positive real numbers, i.e., $$x>0$$.
Range: The range of the function is all real numbers greater than or equal to 6, i.e., $$y\geq 6$$.
Asymptote: The graph has a vertical asymptote at $$x=0$$, which means that the function approaches infinity as $$x$$ approaches 0 from the right.
Intercepts: The graph has a y-intercept at $$(0, 6)$$, which means that the function approaches 6 as $$x$$ approaches 0 from the left.

Q: Can I shift the graph of a logarithmic function horizontally?

A: Yes, you can shift the graph of a logarithmic function horizontally by adding a constant to the x-coordinate of each point on the graph. For example, to shift the graph of $$g(x)=\log_4 x$$ 2 units to the right, you need to add 2 to the x-coordinate of each point on the graph to obtain the graph of $$f(x)=\log_4(x-2)$$.

Q: What are some common mistakes to avoid when working with logarithmic functions and graph transformations?

A: Some common mistakes to avoid when working with logarithmic functions and graph transformations include:

Not checking the domain and range of the function: Make sure to check the domain and range of the function before graphing it.
Not considering the asymptotes: Make sure to consider the asymptotes of the function when graphing it.
Not shifting the graph correctly: Make sure to shift the graph correctly by adding the constant to the y-coordinate or x-coordinate of each point on the graph.